We start by considering a simple production-based model with an AK
production technology subject to adjustment costs. By design, this
formulation has close ties to much of the long-run risk with consumption
exogenously specified consumption endowments. Prior to introducing
climate change, we include two modifications. First, we include an
energy input in way that is mathematically similar to what is used in so
called ``DICE’’ models; and second, we allow for R&D that could
eventually remove the need for energy input. In regards to the second
modification, some suggest that an economically viable version of
nuclear fusion might achieve this aim.
5.1 Production and Innovation
The economic component to our model has two endogenous state
variables: the stock of productive capital and the stock of research
and development capital.
The stock of productive capital, \(K_t,\) evolves as
\[dK_t = K_t \left[ - \mu_k + \left({\frac {I_{t}^k}{K_t}} \right) -
{\frac { \kappa} 2} \left( {\frac {I_{t}^k} {K_t}} \right)^2 \right] dt
+ K_t \sigma_k dW_t\]
where \(\sigma_k\) is a row vector with the same dimension as the
underlying Brownian motion. Capital is broadly conceived to include
human capital and intangible capital. Investment \(I_t^k\)
contributes new capital subject to an adjustment cost captured by the
curvature parameter \(\kappa\).
A process \(J\) captures the stock of knowledge induced by research
and development as measured by \(J_t\):
\[d J_t = - \zeta J_t dt + \psi_0 \left(I_t^j\right)^{\psi_1} \left(J_t\right)^{1 - \psi_1} dt + J_t \sigma_j dW_t\]
where \(0 < \psi_1 < 1\) and \(I_t^j\) is an investment in
research and development (R & D).
While we will solve a social planner’s problem, we will subsequently
entertain the possibility that this evolution equation includes an
externality associated with R&D. For pedagogical simplicity, we consider
the case of a single technology jump to a fully productive green
technology. The parameter, \(\zeta,\) captures depreciation in the
stock of knowledge pertinent for future technological progress.
5.1.1 Output
Output is split between consumption, productive and R&D capital
investments, and emissions abatement expenditure.
\[C_t + I_t^k + I_t^j = \alpha K_t \left[1 - \phi_{0,t}\left(\iota_t\right)^{\phi_1} \right]\]
for \(\phi_1 \ge 2\) and \(0<\phi_{0,t} \le 1,\) where
\[\iota_t = \left(1 - \frac {{\mathcal E}_t}{\beta \alpha K_t} \right){\mathbf 1}_{{\mathcal E}_t < \beta \alpha K_t }\]
and where \({\mathbf 1}\) is an indicator function that assigns one
to the event in the parentheses. When emissions fall short of the
threshold \(\beta_t \alpha K_t,\) there is a corresponding convex
adjustment in the output given by the right-hand side of output
constraint.
This technology is, by design, homogeneous of degree one. For a fixed
\(K_t,\) the implied production function is flat when emissions
exceed the threshold of \(\beta \alpha K_t,\) and has a zero left
derivative at this point. The function equals \(1-\phi_{0,t}\) when
\({\mathcal E}_t=0\) and increases up the threshold as a concave
function with curvature dictated by the parameter \(\phi_1.\) We
feature the case in which \(\phi_1 = 3.\)
from src.plot import plot_climatehist
plot_climatehist("Figure 5: Distorted climate model distribution")
from src.plot import plot_simulatedpath_full2, plot_simulatedpath_uncer_decomp2
title = "Figure 6: Distorted Probability of a Technology Change Jump"
plot_simulatedpath_full2(graph_type="distorted_tech_prob", graph_title = title, yaxis_label="", graph_range=[0,1], before15=False)
from src.plot import plot_simulatedpath_full2, plot_simulatedpath_uncer_decomp2
title = "Figure 7: Distorted Probability of a Damage Jump"
plot_simulatedpath_full2(graph_type="distorted_damage_prob", graph_title = title, yaxis_label="", graph_range=[0,1], before15=False)
from src.plot import plot_gammahist
plot_gammahist("Figure 8: Distorted Climate Model Distribution")
5.2 Social Valuation
The first-order conditions for the socially efficient R & D investment
are:
\[\frac {\partial {\widehat V}}{ \partial j } (X_t) \psi_0 \psi_1 \left( I_t^j \right)^{\psi_1-1} \left( J_t \right)^{1-\psi_1}
- \delta \left(C_t\right)^{-\rho} \left(N_t\right)^{\rho-1}
\left(\exp \left[(\rho - 1) {\widehat V}(X_t)\right] \right) = 0.\]
where we replace \(C_t\) with \(\frac{C_t}{N_t}\) in the
preference specification.
Thus
\[\left( I_t^j \right)^{1 - \psi_1} = \psi_0 \psi_1\left( J_t \right)^{-\psi_1} N_t \left[ \frac { J_t \frac {\partial {\widehat V}}{ \partial j } (X_t) }{
\delta \left(C_t\right)^{-\rho} \left(N_t\right)^{\rho }
\left(\exp \left[(\rho - 1) {\widehat V}(X_t)\right] \right)}\right] .\]
The term in brackets is the social value of the knowledge stock of R & D
expressed in units of (damaged) consumption. We now give an asset
pricing representation of social value of the stock of R & D which is
the intertemporal input into the social benefit for R & D investment.
from src.plot import plot_simulatedpath_full2, plot_simulatedpath_uncer_decomp2
title = "Figure 9: Simulated Pathways of the Log of Social Value of R&D"
plot_simulatedpath_full2(graph_type="LogSVRD_Plot", graph_title = title, yaxis_label="", graph_range=[4,10], before15=True)
from src.plot import plot_simulatedpath_full2, plot_simulatedpath_uncer_decomp2
title = "Figure 9: Simulated Pathways of the Log of Social Value of R&D"
plot_simulatedpath_uncer_decomp2(graph_type="LogSVRD_Plot", graph_title = title, yaxis_label="", graph_range=[4,10], before15=True)
from src.plot import plot_simulatedpath_full2, plot_simulatedpath_uncer_decomp2
title = "Figure 10: Simulated Pathways of the Log of Social Cost of Global Warming"
plot_simulatedpath_full2(graph_type="LogSCGW_Plot", graph_title = title, yaxis_label="", graph_range=[6,14], before15=True)
from src.plot import plot_simulatedpath_full2, plot_simulatedpath_uncer_decomp2
title = "Figure 10: Simulated Pathways of the Log of Social Cost of Global Warming"
plot_simulatedpath_uncer_decomp2(graph_type="LogSCGW_Plot", graph_title = title, yaxis_label="", graph_range=[6,14], before15=True)
from src.plot import plot_simulatedpath_full2, plot_simulatedpath_uncer_decomp2
graph_title = "Figure 11: R&D Investment as Percentage of GDP"
plot_simulatedpath_full2(graph_type="RD_Plot", graph_title = graph_title, yaxis_label="%", graph_range=[0,10], before15=True)
from src.plot import plot_simulatedpath_full2, plot_simulatedpath_uncer_decomp2
graph_title = "Figure 11: R&D Investment as Percentage of GDP"
plot_simulatedpath_uncer_decomp2(graph_type="RD_Plot", graph_title = graph_title, yaxis_label="%", graph_range=[0,10], before15=True)
from src.plot import plot_simulatedpath_full2, plot_simulatedpath_uncer_decomp2
graph_title = "Figure 12: Emissions"
plot_simulatedpath_full2(graph_type="e", graph_title = graph_title, yaxis_label="", graph_range=[6,16], before15=True)
from src.plot import plot_simulatedpath_full2, plot_simulatedpath_uncer_decomp2
graph_title = "Figure 12: Emissions"
plot_simulatedpath_uncer_decomp2(graph_type="e", graph_title = graph_title, yaxis_label="", graph_range=[6,16], before15=True)
5.2 Social Valuation¶
The first-order conditions for the socially efficient R & D investment are:
where we replace \(C_t\) with \(\frac{C_t}{N_t}\) in the preference specification.
Thus
The term in brackets is the social value of the knowledge stock of R & D expressed in units of (damaged) consumption. We now give an asset pricing representation of social value of the stock of R & D which is the intertemporal input into the social benefit for R & D investment.